Antoine Chambert-Loir

IGUSA INTEGRALS AND VOLUME ASYMPTOTICS IN ANALYTIC AND ADELIC GEOMETRY

We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures.

Heights and measures on analytic spaces. A survey of recent results, and some remarks

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related results.

Integral points of bounded height on partial equivariant compactifications of vector groups

We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.

Points of Bounded Height on Equivariant Compactifications of Vector Groups, I

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

Analytic Curves in Algebraic Varieties over Number Fields

We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions on algebraic curves that extends the classical rationality theorems of Borel–Dwork and Polya–Bertrandias, valid over the projective line, to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and p-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces of these arithmetic criteria.

Torseurs Arithmétiques Et Espaces Fibres

We study the compatibility of Manin’s conjecture with natural geometric constructions, like fibrations induced from torsors under linear algebraic groups. The main problem it to understand the variation of metrics from fiber to fiber. For this we introduce the notions of “arithmetic torsors”, “adelic torsion” and “Arakelov L-functions”. We discuss concrete examples, like horospherical varieties and equivariant compactifications of semiabelian varieties. These techniques are applied to prove “going up” and “descent” theorems for height zeta functions on such fibrations.

Points of Bounded Height on Equivariant Compactifications of Vector Groups, II

We prove asymptotic formulas for the number of rational points of bounded height on certain equivariant compactifications of the affine plane.

Structure Theorems for Greenberg Schemes

Throughout this chapter, we denote by R a complete discrete valuation ring with maximal ideal \(\mathfrak{m}\) and residue field k. For every integer n⩾0, we set \(R_{n} = R/\mathfrak{m}^{n+1}\).

Prologue: p-Adic Integration

Motivic integration and some of its applications take they very inspiration from results of p-adic integration, that is, integration on analytic manifolds over non-Archimedean locally compact fields.